Market Risk and Portfolio Theory
General information
References
Prerequisites
Introduction
0.1 Background
The fast rate of innovation that the world has lived since the industrial and the digital revolutions, has only been possible thanks to the existence of a robust financial sector. Without the capital to finance new companies and endeavors (from the construction of train lines to funding the research leading to the personal computer and mobile phones), many of those discoveries would have not been possible or would have taken much longer than they did.
Indeed, let us look at some of the essential roles of the financial system:
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It allows the transfer of economic resources through time and between different people/companies/countries (ex: bonds)
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It provides ways to manage risk (ex: insurance, options)
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It contributes to the flow of economic information and provides means to establish prices (ex: Stock Exchanges)
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It eases trade by providing ways to clear and liquidate payments (ex: FX, credit cards)
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It provides mechanisms to aggregate and divide resources. (ex: shares)
These goals are achieved by exchanges involving financial institutions, financial instruments and financial markets:
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Financial institutions are the agents whose main purpose is to provide financial services and are therefore the main interacting agents (ex: banks, insurers, regulators,…).
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Financial instruments are the assets that are exchanged. They belong to three main classes: debts, equity, derivatives.
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Financial markets are the ”places” where those exchanges happen and the ”rules” governing those exchanges 11 1 The“actual” way markets ”exist” is very involved and has a lot to do with computers, communications, algorithms and regulations. For an interesting way of having a partial view of equity and derivative markets in the US, read Michael Lewis’ Flash Boys.
As pointed out above, risk is an intrinsic element of financial markets and, in fact, of every business activity. Let us remind ourselves what this notion means. The Oxford English dictionary offers the following definition
Definition 0.1.
By risk we understand the possibility of financial loss or failure as a quantifiable factor in evaluating the potential profit in a commercial enterprise or investment.
We are interested in understanding the interaction of market participants with risks. In particular, we will focus on market risk, that is
Definition 0.2.
Market risk is the type of risk associated to the uncertainty of values of financial instruments in financial markets.
For example, a bank would qualify as market risk the following risks:
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Default risk, interest rate risk, credit spread risk, equity risk, foreign exchange risk and commodities risk for trading book instruments; and
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Foreign exchange risk and commodities risk for banking book instruments
To illustrate what market risk means, we can look at Figure 1: it highlights the fall in the FTSE100 index around Black Monday (19.10.1987), where around 27% of market value of this index was lost in one day. The index would take more than one year to retrieve its previous value.
We have explicitly asked in our definition for risks to be quantifiable. We reserve another word uncertainty to potential losses that are not quantifiable. This is a very important property, because it means we can measure it. This will allow us to define risk measures.
Another consequence of being able to quantify risks is that it allows for mathematical modelling: that is, to find a simplified representation of our object of study, in mathematical terms, with the aim of better understanding it and answer questions related to it.
In this course, we are interested in studying mathematical models of financial markets, with the aim to understand the effects of market risk, in particular those related to decision making. To illustrate this idea, let us give some of examples of the main questions we will try to understand
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How do we quantify risks? Can we do something about them?
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How to measure the performance of an investment? Are portfolios that are optimal in some performance sense?
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How to decide if we should access a financial instrument? For example, when should someone buy an insurance, or ask for a credit? More in general, is there a way to make optimal decisions?
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What should be the price of newly introduced financial assets?
Financial markets have evolved to be very complex entities comprising thousands of market participants of different types (banks, hedge funds, pension funds, insurers, …), dealing with many types of instruments (stocks, bonds, FX, …) through platforms where precise rules for matching operations and dealing with settlement requirements and regulations that can change from one country to the next. To obtain meaningful quantitative conclusions on the questions we raised, we introduce market models, a simplified mathematical description of the participants of the market, the instruments available and the rules under which they can trade them.
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We do not want to study specific market participants instruments and institutions. We rather create a (mathematical) abstract model that represents them all. In particular we do not want to explain why the values of a financial instrument are risky, but rather try to model their “unpredictable” changes. We are also interested in understanding the attitudes of participants.
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As with any model, we try to find a good balance between complexity and tractability: our purpose is to make simplifying assumptions that keep the results meaningful. In any case we must always remember that any results are obtained under the model, and have to be taken understanding its limitations. In many occasions the insights they produce are more important that their exact values.
Chapter 1 Fundamentals of market theory
What properties characterise a “good” market model?
In this chapter we consider the mathematical modelling of a financial market in discrete time with a finite horizon . The market is composed of assets that can be freely bought or sold. We model the asset total values as stochastic processes in a filtered probability space. One of the assets (identified by the index 0) is assumed to be locally risk-free.
1.1 A market model in discrete time
We assume that we are given a probability space , where is interpreted as the set of all possible outcomes in the market, is the sigma algebra of all measurable events and is the probability quantifying the likeliness of those events11 1 See the lecture notes by Terence Tao on the subject (terrytao.wordpress.com/category/teaching/275a-probability-theory/) for a good refreshment in probability theory..
Let us introduce our characterisation of financial instruments in the market:
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We identify each asset by a number . The number 0 will be reserved for a bank account asset (see Definition 1.7 below).
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Each asset is completely characterised by a random process (a random vector in the finite horizon case) that represents the total market value of the given asset. These values are expressed with respect to a common numéraire or currency. Moreover, we assume that the actions of a single agent on the market cannot by themselves change these values.
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We assume that investors can buy as many shares/instances of an asset as they want, in fractional quantities if they so desire. They can also short an asset as much as they want.
For convenience, we introduce also the following matrix-like notation: we write , that is, a column vector with the random values at time of each asset.
1.1.1 Structure of the market
An important part of a multi-period financial market is the information that is available at each time period. It then makes to add some structure to our probability space to include the flow of this information: we add a filtration, that is a sequence of sigma algebras , indexed by (typically either , or a finite interval starting at zero), that is increasing in the sense of inclusion , i.e.
Intuitively, the filtration at time contains all the events that are decidable by time . For example, assume that the event “The value of the asset at time 2 is larger than £100” (which we would write as the set of outcomes for which ), is something we know at time 2 and afterwards, so it belongs to for all .
In the following, unless otherwise stated, we consider only filtrations defined over the set . This corresponds to the finite time discrete model case. Additionally, we make the following assumption
Assumption 1.1.
The filtration satisfies
The above assumption simply says that -measurable functions are deterministic and in particular that the initial price of all assets is known. It also says that the whole set of outcomes is known by time , which is a convenience assumption that signifies that the model ends at that time.
It is important to distinguish processes which are compatible with the information structure given by the filtration.
Definition 1.2.
We say that a stochastic process is adapted, if is measurable for all . We say that a stochastic process is predictable, if is measurable for all .
In our setup, the property of being adapted captures the intuition that processes are progressively discovered and they should not anticipate the future; likewise, the property of being predicable captures variables whose value at the end of a period is known at the beginning of it. Clearly, by the increasing nature of filtrations, any predictable process is adapted. It is then natural to make the following assumption
Assumption 1.3.
The value processes are adapted to the filtration .
This means that we assume that values at time are revealed at that time, and those values are remembered ever since.
Conditional expectation
In probability, the expectation is an operator that gives us a punctual estimator on a random variable. Now, as information is evolving, the likeliness of events (measured by the probability) also evolves. Thus, what we deem the best punctual estimator now (at time 0, the start of the model) does not coincide with the best estimator at time given that we have more information about what actually happened (and what did not) at the times before . We capture this feature with the concept of conditional probability.
Definition 1.4.
The conditional expectation at time (denoted by , or if the filtration is clear from the context) is an operator such that for each measurable random variable with finite variance assigns an - measurable random variable given by,
| (1.1) |
In the above, we call to the set of all random variables that are measurable and such that . Hence, conditional expectation is the -measurable random variable that best
approximates in a mean quadratic sense. The operator can be extended to random variables with finite mean.
We give without proof a very useful characterisation of conditional expectations.
Proposition 1.5.
Given a random variable such that , the conditional expectation is the only (up to measure zero) -measurable, integrable random variable such that
| (1.2) |
The above characterisation is handy whenever we want to verify a given candidate to be a conditional expectation.
Some properties of conditional expectation
We list now some important properties of conditional expectation. The proof of them is easily deduced either from the characterisation (1.5) or from using similar properties of the expectation operator in Definition 1.1. Let be measurable.
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(Invariance): If is -measurable,
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(Homogeneity): If is -measurable,
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(Orthogonality): If is independent of ,
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(Tower property): If , .
Moreover, as for the expectation operator we have
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(Linearity): We have for all
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(Positivity): if , (with linearity, we get monotonicity…)
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(Independence): If , .
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Monotone convergence: , then
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Dominated convergence: and with integrable then
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Jensen’s inequality: If is convex, then
Example 1.6.
Assume that is i.i.d, with each entry being integrable, and assume we are working under the filtration generated by . Assume also that . For all ,
Intuitively, conditional expectation is “our best estimator” of a random variable given that “we know everything that happened up to ”. For this reason, it will play an important role when considering prices of contingent claims, as will be seen in what follows.
Bank account
As announced before, the asset will play a special role. A very convenient assumption is that it is a locally risk-free asset. It is customary to call such an asset bank account or money market account.
Definition 1.7.
We call money market, bank account or (locally) risk-free asset to an asset whose price is strictly positive and predictable (see definition 1.2) with . We call any asset whose price process is not predictable risky
Hence, the bank account is an asset that can be overall random, but such that at each period, we have certainty on its value by the end of the period already at the start of the period. Observe, in particular, that if , the bank account is deterministic: in this context, it is also known as a risk-free asset. The assumption that is a normalisation and is useful to simplify some expressions.
In practice, some assets may act close to a money market account, like government bonds or overnight collateralised swaps. We will not systematically assume its existence, but it will be clear from the discussion that many developments are enhanced or simplified when one is available.
A money market account is a natural benchmark for all other assets. We will argue in the following chapters that investors usually require a larger average compensation in order to assume risk. This excess of mean return is denoted risk premium.
Definition 1.8.
The (conditional) risk premium of asset in the period is defined as
| (1.3) |
Sometimes, the risk premium is also called mean excess return.
Discounted prices
It is often convenient to express prices relative to the money account, i.e.
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where we use the fact that this process is strictly positive. In financial terms, we say that is a discounted price. Note that this is equivalent to using as a numéraire.
Returns
Because we have assumed that an investor can buy any quantity of a given asset, it is possible and sometimes more convenient to focus on how prices change rather than what the actual prices are. This motivates the following definitions
Definition 1.9.
The (gross) return on asset for the period for is defined as
| (1.5) |
whenever the ratio is well-defined. Clearly, returns convey information in the case when total values are positive, so they are particularly useful when dealing with assets like stock prices.
Definition 1.10.
The rate of return (or net return) on asset is defined as
| (1.6) |
For example, if an investor decides to invest an amount in asset at time , then we can get directly the amount they will have at the end of the first period by multiplying by the gross rate since
where represents the number of shares that can be bought with .
Likewise, investors can evaluate their net gain or loss (their actual earnings through the investment) using the rate of return, since
Note that since it is possible to buy any fraction of a stock, for all modelling effects, two assets with the same gross return are indistinguishable.
We can also define a discounted net gain of an investment, which would be given by
| (1.7) |
1.2 Some examples of market models
To illustrate our discussion up to now, we present here some well known market models in discrete time.
1.2.1 Independent and identically distributed returns
In this simple model, all returns in different periods of each asset are all assumed to be identically distributed and independent from each other. Moreover, the filtration is taken as the generated by the return process, so that for all and all
Under this assumption, and knowing the initial value of each asset, the value at time is found to be
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On the other hand, we find that for all
| (1.9) |
Indeed, note that using the tower and independence properties of conditional expectation, and the i.i.d. fact, we get
A log-normal model with deterministic money market account
A quite popular instance of the i.i.d. model postulates that the money market account is deterministic (i.e. ) for some strictly positive function (for example for a fixed ), and that the returns of the other assets follow a log normal distribution, with
where denotes the vector of the assets , is a vector of means and is the variance-covariance matrix.
The advantage of the log-normal assumption is that we get from (1.8) that
which, thanks to the linearity invariance property of Gaussian vectors, is also log normal, but with mean and variance rescaled by a factor of .
The continuous version of this process is the well known Merton-Black-Scholes model.
The Cox, Ross and Rubinstein model (Binomial model)
Another, instance of the i.i.d. model that is usually applied in the case , keeps the deterministic money market account, and postulates that , where are binomial random variables with probability .
If , this model can be defined on a finite probability space , with so that . The value of the total stock value would correspond to the following binomial tree depicted in (1.1).
Note that case might require a larger probability space to implement, depending on the joint structure between the different assets. If all assets are assumed to evolve independently, one can construct the model over the space with a well defined probability structure.
1.2.2 Markovian models
The i.i.d. assumption is very strong: it means that returns do not evolve over time. We can allow for changes in the returns but keeping one common trait: that prices are “memory-less” in the sense that the dynamics of the process after a time only depend on its state at that time and not on the path the process took to arrive to it.
Let us formalise this intuition. Recall that for any two random variables with finite expectation, we can define the conditional expectation to be a random variable satisfying:
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There exists a measurable such that almost surely
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, for any measurable function .
As expected, satisfies analogous properties to the invariance and orthogonality properties of conditional expectation with respect to a filtration. Note that, in the particular case where has a joint density (say ), we have that , where
We are ready to define Markov processes.
Definition 1.11.
X is a Markov process if for every measurable function and ,
for some measurable function .
By definition, the i.i.d models before are all Markovian. Let us give a couple of examples of Markovian models for which returns are not i.i.d..
Auto regressive models of order 1 (AR(1)) with one factor
This model over the total value vector assumes that the are defined by
with the being i.i.d, and known for each . We assume the filtration is the one generated by . Note that neither nor are i.i.d. in this case. However, we have
Hence, by induction, we get for all that
which clearly shows that this model is Markovian (the function in the Definition (1.11) is simply linear).
A generalisation of this model to several factors is straightforward.
General 1-factor Markovian model
For this model over the total value vector , we choose measurable functions and assume that the are defined by . with the being i.i.d, and known for each . As before, we assume the filtration is the one generated by . In this case,
however, using property i of conditional expectation, we get that
for some measurable functions . An induction argument as in the auto-regressive case then shows that this model is Markovian. Once again, this model can be easily generalised to several factors.
1.2.3 Non-Markov models
Let us see some examples of possible non-Markovian models (a.k.a. path-dependent processes). Once more, we assume the filtration is th eone generated by the process , which is assumed to be i.i.d.
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Auto-regressive of second order (AR(2)): defined by .
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Mild support and resistance process: defined by
where is the positive part function.
In many cases, as the ones above, it is possible to render the model Markovian by extending the set of state variables, that is, we provide a model for the vector of total value prices and for some extra variables (for example, the process extended by the delayed prices).
Exercise.
Choose a set of state variables to enlarge the set of state variables to render the above examples Markovian.
1.3 Arbitrage and completeness
We are ready to introduce the set of actions that a market participant can undertake on our market model and characterise some good properties of the market with respect to these actions. We start with a definition.
Definition 1.12 (Strategy).
A strategy is a set of actions that an investor decides to perform on the market (in order to obtain their goal). It is a predictable process with values in so that denotes the number of shares of asset to be held during the period .
The predictable property is imposed to reflect the fact that investors decide on their strategy with the information available to them at the start of each period.
Remark 1.1.
In the one-period model, the only choices available to investors are at initial time. Hence in the one-period model, choosing a strategy means simply choosing a portfolio composition, i.e. a vector .
From the definition, we see that the amount invested on the asset at time is . At the end of the period, just before recomposition, the investor would have as a result .
We denote by the value of a portfolio that follows the strategy . By definition,
and by time , before any changes in the composition (that we denote ), this would become
If investors only put forward some money at initial time, and later only redistribute the value of their portfolios, we say that the strategy is self-financing. In our model, this reads as follows
Definition 1.13 (Self-financing strategies).
A strategy is self-financing if for all ,
Exercise.
Show that a strategy is self-financing if and only if for all ,
We say that there is an arbitrage opportunity if we can construct a costless self-financing strategy that produces some returns but never losses by the end of the modelling time. Let us make this definition precise:
Definition 1.14.
A market allows for an arbitrage opportunity, if there exists a self-financing strategy such that
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A “good” market does not allow arbitrage opportunities to exist: indeed, an arbitrage opportunity would be rapidly detected and eliminated as a consequence of investors exploiting it. Thus, arbitrage opportunities are inefficiencies to be exploited, while a no-arbitrage condition intuitively implies that all investors have a good access to the market and to information.
We call a market model where arbitrages are not possible arbitrage-free.
Another characteristic of a “good” market model is for it to be rich enough so that one can reproduce, at time any desired wealth profile.
Definition 1.15.
We say that a wealth profile (at time ) can be replicated if there exists a self-financing strategy such that
The replication price is in this case simply the initial investment required to acquire such a self-financing strategy, i.e. . If such a strategy exists , we also say that replicates .
Definition 1.16.
We say that a market is complete if every -measurable wealth with and can be replicated.
Since strategies are so limited in one-period models, market completeness is a very strong assumption for this type of model: in essence, we need to have at least as many assets as different possible outcomes on the probability space. As more opportunities to rebalance a portfolio are given (i.e. when we move to multiperiod models), it becomes easier to replicate a given profile, or equivalently, we require less assets to guarantee completeness. In some continuous models (where constant rebalancing of a portfolio is allowed) complete models are possible even when only one risky and a risk-free asset are available.
A complete market is a well developed market, as there is an incentive for financial institutions to introduce new assets to the market to account for uncovered profiles. However, in practice, completeness is limited by aspects like regulation, moral hazard, adverse selection and sophistication aversion. An important criterion to choose the price of a new asset, is that it would not introduce an arbitrage in an arbitrage-free market: indeed, if this were the case, either the financial institution introducing the asset or their clients could face certain losses. This motivates the following definition
Definition 1.17.
Consider an arbitrage-free market model. Assume that a contingent claim (i.e. an asset whose price depends on the value of other assets) with pay-off at time wants to be introduced to the market. We say that is an arbitrage-free price if the market extended with the new asset at the new price is still arbitrage-free.
Finding arbitrage-free prices is an important task in financial mathematics. In several cases, the problem can be solved by introducing some structural characteristics of a arbitrage-free market: either the existence of a special process, the stochastic discount factor or the introduction of a new measure the risk neutral measure. We study these in the following sections.
1.4 Stochastic discount factor (SDF)
In a “good” market model (in the sense of having the rules for buying assets and the no-arbitrage and completeness properties we have presented), there are some structural relationships between the asset values at the beginning and at the end of each period: for example, given that we know the random variable of the value of an asset at the end of the period, we know (by no arbitrage) that some initial prices would be ruled out. We will see in the following that the relationship is even stronger. We need to introduce some new concepts.
Definition 1.18.
We say that an adapted process in is a stochastic discount factor (SDF) process, if we have that , and for each
| (1.10) | |||
| (1.11) |
The intuition behind this definition is as follows: if we assume that the market is assumed to be risk neutral (i.e. pricing by taking only expectation), the current prices are explained if we suppose that the market discounts pay-offs depending on both the time of occurrence and the actual event that arises. This is encapsulated in the random process . Indeed, it follows from the fact that and (1.11) that for any ,
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The SDF can also be seen as connecting returns. Indeed, by dividing by in (1.11) we get
| (1.13) |
and by the properties of conditional expectation,
which implies that the SDF is orthogonal to relative excess returns.
Exercise.
Using induction prove that for all ,
| (1.14) |
Example 1.19.
Let us look at an instance of the binomial model on returns presented in section (1.2.1). We assume there is one deterministic asset and two risky assets in a multiperiod market model with periods. More specifically, we assume that with uniform probability and
Where we assume that , , and . Now, let us set
with
Note that , due to our constraints on the constants of th problem. We now show that
is a strictly positive stochastic discount factor for this market model.
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We first verify the initial condition. Indeed, .
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We now verify that . However, this is trivial as long as since we are in a finite probability space.
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Finally, we check the property of conditional expectations. We get
Hence, is indeed an SDF. Finally, note that if we add the condition
the SDF defined above is strictly positive. ∎
Martingales
The structure of the processes is very special: the best estimation at time for the future values of the process is precisely the value of the process at time . Such processes are called martingales.
Definition 1.20.
An integrable and adapted process is a martingale if
i.e., if the best -measurable approximation of the process at
every time bigger that is its value at itself.
Example 1.21.
The definition of an SDF guarantees that (the total market value of each asset weighted by the stochastic discount factor) are martingales.
The above example can be made stronger.
Proposition 1.22.
An adapted process is a stochastic discount factor if and only if the process is a martingale for any bounded self-financing strategy .
Exercise.
Prove Proposition 1.22.
Stochastic discount factors are not guaranteed to exist or to be unique. However, some remarkable results connect the existence and uniqueness of an SDF with the no-arbitrage and completeness market properties. These theorems are the so-called fundamental theorems of asset pricing. In our simple discrete-time finite world, they read as follows:
Theorem 1.23 (First fundamental theorem asset pricing).
A market has no arbitrage opportunities if and only if there exists a (strictly) positive SDF.
Theorem 1.24 (Second fundamental theorem asset pricing).
An arbitrage-free market model is complete if and only if there is a unique SDF.
We present the proof of these theorems in the particular case of finite probability space in the one-period model in section 1.6.3 below.
1.5 Risk neutral probabilities
As we explained before, assuming the market is risk-neutral, the introduction of an SDF accounted intuitively for our observed prices by introducing a discounting term. An alternative is to assume that the market has a different perception of the probabilities associated to each possible outcome. The risk neutral property will be expressed in the fact that under this alternative probability measure (that in the following we denote ), the conditional risk premium of any asset is zero.
Let us start by reminding ourselves the concept of equivalent measures.
Definition 1.25.
We say that the probability measure is absolutely continuous with respect to the probability measure (denoted ), if assigns zero probability to any set with zero probability under (i.e. ). If both and , we say the two measures are equivalent (denoted ).
Thanks to the Radon-Nikodym theorem (see for example Durrett (2010)), we know that equivalent measures can be shown to be connected via a positive random variable with unit expectation that re-weights the observations and is called a density. In the following we show how to construct the risk neutral measure by defining appropriately its density.
Let us assume that the market is arbitrage-free. We know that in this case there is a strictly positive SDF (). We can then define the random variable
| (1.15) |
The expectation operator of the risk neutral probability for any random variable is then given by
| (1.16) |
Equivalently, we can define for each event by,
| (1.17) |
Let us verify that actually is a density.
Proposition 1.26.
If is a strictly positive SDF, the operator defined above is a true probability and .
Proof.
The claim follows from the fact that is strictly positive by the assumptions on and . Indeed, we can verify that so defined is a true probability. Indeed:
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Since is strictly positive, trivially
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Set pairwise disjoint, we get
Finally, we also deduce that . Hence both probabilities are equivalent and is the Radon Nikodym density almost surely. ∎
Lemma 1.27.
Let , an arbitrary random variable and and defined as above. Then,
Proof.
Definition 1.28 (Risk-neutral measure).
We say that a measure defined on is a risk-neutral measure if it is equivalent to and its Radon-Nikodym density can be written as in (1.15) for some strictly positive stochastic discount factor .
Remark 1.2.
From the definition, we can deduce that there is a one-to-one correspondence between risk-neutral probabilities and strictly positive stochastic discount factors. As a consequence, arbitrage-free markets might have more than one risk neutral probability with uniqueness if and only if the market is complete.
The property of a process of being a martingale depends both on the filtration and the probability used to evaluate conditional expectations. Sometimes it will be convenient to evaluate the property with respect to a measure different than . The following theorem makes such a claim for a process when considering conditional expectations with respect to . It is a very well known theorem in financial mathematics.
Theorem 1.29.
A measure is a risk-neutral probability if and only if is equivalent to and discounted prices of market instruments (see (1.4)) are -martingales.
Proof.
Assume first that defined by the density (1.15). We have already verified that this implies that is equivalent to . It remains to show that discounted prices are martingales. We check first that
| (1.18) |
by the definition of stochastic discount factor.
Now, recall from (1.14) that the process is a martingale. Set . Then, we have from Lemma 1.27 and the tower property of conditional expectation that
| (1.19) |
Consider now the opposite direction, and assume that is an equivalent measure under which discounted prices are martingales. Let us define for all
Note that trivially, by setting , . We just need to check that such a process would be a strictly positive SDF. But this follows directly from (1.18) and (1.19). ∎
Due to Theorem 1.29, risk-neutral measures are also frequently called martingale measures.
Let us close this section by verifying our claim that under the risk neutral measure there are no risk premia. We prove first the following small lemma
Lemma 1.30.
Assume is a money market account. For all and any -measurable random variable
Proof.
We can now deduce the following result.
Proposition 1.31.
The conditional risk premium under the risk neutral probability of any market asset on any period is zero.
1.6 A deeper look at one-period models in a finite probability space
In order to better illustrate the theory we have introduced up to now, let us consider in this section the case of market models in one period () where the possible set of outcomes is finite, i.e. where for some . We take the complete sigma algebra (i.e. we can find the probability of every subset of ). Recall that in the finite case, the probability measure is completely determined by the probability of each singleton . We assume they are all different from zero (otherwise we reduce the number of states).
There are two main advantages of studying the finite probability setting: the first one is that due to its simplicity, we will be able to give a further economic interpretation to the SDF. Secondly, we will be able to rewrite many properties of the market in terms of matrix operations. Hence, we are able to use results in linear algebra to verify easily market properties like completeness or no-arbitrage, and show simply the F.T.A.Ps.
1.6.1 Understanding the SDF in terms of Arrow-Debreu securities
Let us assume that the market is arbitrage-free and complete. In this case, we can get a better intuition of what an SDF means. We introduce the concept of an Arrow-Debreu security, as a security that identifies outcomes on the probability space.
Definition 1.32.
A security that pays one unit of the consumption good in a particular state , and pays zero in all other states is called an Arrow security or an Arrow-Debreu security. In mathematical notation, the AD security is written .
Let be the price of the AD security corresponding to state . Informally, we can understand their price as the value in present time of guaranteeing a unit in a given scenario. Because of market completeness, we can replicate each AD security with market instruments and there is only one price for each market instrument. Hence,
| (1.20) |
or equivalently,
| (1.21) |
which can be understood as a re-weighting of the probability by the price of a unit on each state. For this reason, is also called “state-price density” or “pricing kernel.”
Remark 1.3.
Note that although we assumed market completeness to explain the notion of Arrow-Debreu securities, the AD prices can be directly found from (1.20), and thus they only require the existence of an SDF. Hence, there is a one-to-one correspondence between AD prices, SDFs and risk neutral probabilities in the finite probability space case.
Finally, note that AD prices can also be used to price an asset, since
| (1.22) |
1.6.2 Matrix interpretation
In the finite state model, we can also rewrite the market properties in terms of vectors and matrices. For example, we introduce the matrix to be given by
which we write succinctly by . Let us also set22 2 We use as has already been taken . Note that each one of the random variables is included as a row vector in . Hence, the vector of expectations for each asset can be found by operating matrix multiplication with the vector on the right, i.e.
while portfolios can be found by operating by the transpose of a vector on the left, i.e.
This last property immediately implies the following characterisation
Proposition 1.33.
A one-period market model in a finite probability space is complete if and only if the range of is .
Moreover, a vector of AD prices denoted would satisfy, thanks to (1.22)
| (1.23) |
Hence we conclude from the observation in Remark 1.3 and (1.20) and (accepting for now) the first theorem of asset pricing 1.23, that
Proposition 1.34.
A one-period market model in a finite probability space is arbitrage-free if and only if there is a strictly positive solution to (1.23).
All in all, this means that we can check easily the main properties of the market and obtain its main descriptors using properties of the matrix .
Example 1.35.
Take a market with two assets and three equally likely scenarios. Assume that and
Note that this market model does not contain any risk-free asset. We can check that this market model is incomplete and admits an arbitrage:
-
•
Since the range of is at most (the number of outcomes), the market is incomplete. For example cannot be replicated.
-
•
The market admits an arbitrage. To check it, we look at equation (1.23). To solve the linear system we extend the matrix and perform matrix reduction operations to get
i.e. , and . Since the last equation cannot have a strictly positive solution, we conclude the market admits an arbitrage.
Alternatively, one could explicitly propose an arbitrage strategy. For example, take . Clearly it has not cost at the start time, produces zero outcome in case that occurs, and positive outcome if either or occur.
1.6.3 Proof of the F.T.A.P.
In this section, we prove the two fundamental theorems of asset pricing in the case when the probability space is finite. In Example (1.35) we used an extended matrix to study the no-arbitrage condition. Inspired by this, we define
Note that we have changed the signs of the initial values of . This will be useful to take profit of separation theorems in the following.
An interesting result that will help us prove the F.T.A.P. is the following alternative theorem. It essentially says that we can either find a point in the intersection of a linear system and a hypercube , or instead finding a vector that is perpendicular to all and that points ”within’ .
Theorem 1.36 (Rockafellar (1970)).
Let be a subspace of , and let be a possibly degenerate but non-empty hypercube i.e. where each is an interval not necessarily bounded and possibly only a point. Then one and only one of the following alternatives holds:
-
•
There exists a vector
-
•
There exists a vector such that for all and for all
We will not reproduce the proof, but rather give an intuitive explanation: if there is no point in the intersection of and , then we can divide the space in two around and will be contained in one of the two parts. In particular cannot be the whole space, and cannot be unbounded in all directions. If we take any element of , this point can be expressed as the sum of one element of and one perpendicular to . One can then prove that the perpendicular component will have the desired property of pointing “toward” .
We can deduce two easy corollaries that will be useful in the following an illustrate the uses of the Theorem.
Corollary 1.37.
Let . If one can show that there is no solution to then there exists , such that , for all .
Proof.
In Theorem 1.36, set , be the image of the matrix , and .
We have that there is a vector such that for all and for all . But this later inequality implies that . Indeed if a component would be such that , one could choose (only one in the direction ‘’ ) and obtain a contradiction.
∎
Corollary 1.38 (A version of Farkas’ lemma).
Let . If one can show that there is no solution to then there exists , such that , for all , and .
First F.T.A.P.
Proof of Theorem 1.23 in finite probability space case.
Claim: No arbitrage there exists a strictly positive SDF.
””: We are going to show that if a strictly positive SDF exists, any strategy that generates no losses and some gains must have a strictly positive initial cost.
Let be a strictly positive S.D.F. Let be a portfolio such that and, for some , . Then , by definition of , we have that, for each .
| (1.24) |
By linearity, we can then deduce that
Hence, the initial cost of must be positive.
””: Let us remark first that no-arbitrage implies that there is no solution to . Then, by Corollary (1.37) (with ), we can find a strictly positive vector such that for all . Hence,
where we can divide using the strict positivity of . To finish, take . ∎
Second F.T.A.P.
Proof of Theorem 1.24 in finite probability space case.
Claim: Arbitrage-free market is complete unique SDF.
””:Consider the Arrow-Debreu securities. Since the market is complete, for each there exists such that . But since the market is arbitrage-free, the price of satisfies .
Now, assume there are two SDFs , . Then, they satisfy for all , so they are equal.
””: Assume the market model is not complete. Then, there exists such that . By Corollary 1.38 we can find a non-trivial such that for any , . Without loss of generality, we pick such that for some , .
Now, let be a vector of AD prices. (we know it exists thanks to the first F.T.A.P.) and (1.3). Define
Note that . Set We have , , and for all ,
| (1.25) | ||||
| (1.26) | ||||
| (1.27) |
Hence, we have two different sets of possible Arrow-Debreu prices. By remark (1.3), the claim follows.
∎
1.7 Exercises
Exercise 1.1.
Let be an i.i.d. discrete time stochastic process and consider the filtration associated to it. For each of the following examples, establish if they are: Predictable, Adapted, Markovian or Martingales (see definitions 1.2, 1.11 and 1.20). Justify your answer by proving or giving a counterexample.
-
1.
and for
-
2.
and for
-
3.
is in addition a Gaussian process (that is, every is jointly Gaussian ), and for
-
4.
and for and ,
-
5.
-
6.
-
7.
Exercise 1.2.
Consider a market model with and , given by a generalised binomial market model, where and for , but where the returns at different times are not necessarily i.i.d.
Let and define and by
-
1.
Can and be interpreted as valid strategies? Justify your answer.
-
2.
Construct an explicit instance of a generalised binomial model where , where this is taken to mean for (almost) all .
-
3.
Give an explicit instance of a generalised binomial model where .
-
4.
Can you construct an instance of the model where and ?
Exercise 1.3.
Consider a multiperiod market model where is a money market account and is a stochastic discount factor. Show that for all ; :
| (1.28) |
Note: The previous exercise shows that when stochastic discount factor exists, the conditional risk premium is determined by the covariance of the asset and the geometrical increment of the SDF. Note also, that the sign structure is very interesting. In general, as we expect risk premia and risk-free returns to be positive, we expect to see a negative correlation between the SDF and the return of market instruments.
Exercise 1.4.
Consider a one-period model, and set , and assume that is such that for . Assume we have three assets with no pay-off and such that
and
-
•
Is there any risk-less asset in this market? If so, what is the risk-free rate?
-
•
Calculate the return and risk premia of the risky assets (Assume the probability law is uniform).
-
•
Is the market complete? Arbitrage free? Justify your answers.
Exercise 1.5.
Still on a one-period model, set , and assume that there are two assets, a risk-free asset with return and a risky asset. Let us call and , and assume that .
-
1.
Find conditions on equivalent to the absence of arbitrage.
-
2.
Compute the risk neutral probabilities of each state
-
3.
Assuming these conditions hold, compute the unique vector of Arrow-Debreu prices
-
4.
Suppose that a call option is introduced in the market: it pays , for some . Assuming that the price of the risky asset is , compute the price of the call option for if the assumptions found before hold.
Exercise 1.6.
We say that a market has the law of one price if for any two portfolios represented by and such that , we have .
-
•
Show that an arbitrage-free market has the law of one price.
-
•
Give an example of a market that admits arbitrage but such that the law of one price is satisfied.
Exercise 1.7.
Consider a one-period market model. Let be two strictly positive SDFs. Show that any convex combination of the two is also a strictly positive SDF, i.e. show that
is also a strictly positive SDF.
Note: We are showing that the set of SDFs is a convex set.
Exercise 1.8.
Show that a price is arbitrage-free for an asset with final value (see Definition 1.17) if and only if for some strictly positive SDF ().
Exercise 1.9.
Show that if are two arbitrage-free prices for an asset, then any convex combination of the two prices is also an arbitrage free price.
Remark 1.4.
The last two exercises show that in an incomplete market, it is possible to find more than one arbitrage-free price, but that there is a structure for the set of arbitrage-free prices: it is an interval (because you showed that any intermediate price between two valid prices is also valid). This is useful when dealing with incomplete markets and is the basis for either robust pricing and minimal super-hedging.
Chapter 2 Utility functions
How do market participants behave when their goal is their well-being ? What are their attitudes with respect to risk?
2.1 Utility functions to model behaviour
Now that we have presented an abstract theory on how to model the market, we can proceed to model investor’s decisions and their connection to risk.
A very important idea that underlies such an effort is the assumption that the choices that investors make are not completely arbitrary or random, but rather that they can be modelled as consequences of the information at hand. This is an old idea, but got a quantitative taste in the 18th century, where the concept of utility as a measurement or degree of “well-being” or “pleasure” was proposed and develop by, among others, Hume and Bentham11 1 Yes, the Bentham whose auto-icon can be found on the South Cloisters here at UCL. This concept, although highly controversial from a philosophical point of view, has been very successful and useful in economics, in particular supporting the development of a whole area now known as microeconomics.
An additional important element in the utility theory is its connection to uncertainty. Probably the first quantitative connection between the two concepts is found in D. Bernoulli’s solution to the famous “St. Petersburg Paradox” 22 2 The original paper can be found translated from Latin in Bernoulli (1954) . For a quick review, check https://en.wikipedia.org/wiki/St._Petersburg_paradox proposed in 1738 in a paper called Exposition of a new theory on the measurement of risk, where the concept of expected utility first appeared. This became a well founded theory based on a rationality assumption and a simple set of rules thanks to the axiomatic development of Von Neumann and Morgenstern in 1947.
However, the arrival of clinical behavioural studies showed several cracks on the actual capacity of utility theory to act as a descriptive theory : individuals were not consistent in their choices, and there were non-rational biases related to, for example, framing and relative risk aversion, that were observed in empirical studies. This lead to a revision of the paradigm initiated by the Prospect Theory of Kahneman and Tverski. The works of other Nobel laureates like Schiller also questioned rationality on the markets. A branch of Behavioural Economics has developed ever since.
Even with its flaws, utility theory and the rational assumptions remain relevant. On the one hand, they provide a natural reference or benchmark: for example, one characterises exuberance by comparison with a rational market. Hence, utility theory can be used to study a model where everything is as it should be, an use it to characterise deviations from it seen in practice. This means that the theory has a normative value. A second observation is that even if the simple utility theory is flawed, the essential idea of being able to capture an investor’s (and in general people’s) preferences in a mathematical way is more alive than ever: the changes brought by Behavioural Economics entailed only modifications of the model, not a fundamental reconsideration on the idea that decisions can be modelled. This notion, that people’s decisions can be characterised, lies behind the extraordinary revolution in ’client knowledge’ and ’prediction analysis’ driven by machine learning, supported by the large amount of data gathered in our ever connected world and used by companies and governments alike to push for their interests.
All the above makes it relevant to understand and study the simple utility function paradigm in one period. The idea of utility function means essentially that investors establish their preferences by using a function that measures their satisfaction and drives their choices.
Mathematically, in a one-period framework, we define a utility function as follows:
Definition 2.1.
A utility function (in one-period) is an increasing continuous mapping , with an interval in .
Utility functions encode our satisfaction or well-being. This is why it is understood as an increasing function: well-being should not decrease when wealth increases (in other words, we expect that more should feel better than less).
Example 2.2.
Here are some examples of utility functions: (exponential, defined over ) , (logarithmic, defined over ).
What happens when a market participant faces an uncertain bet? The key proposition of Bernoulli is that investors are sensitive to their expected well-being Hence, among two possible investments, an investor will choose the one with the biggest expected utility. That is, when given a choice between two different random investments and ,
If both investments share the same expected utility they are said to be equivalent.
2.2 Risk attitude and certainty equivalence
One of the consequences of modelling the preferences of an investor as being ruled by maximising expected utility is that we can understand how they react to risk: for example, we can understand if they fear or look for risk.
Definition 2.3.
An investor is said to be “weakly” risk averse if
| (2.1) |
It is called “weakly” risk seeking if
| (2.2) |
In other words, a risk averse investor prefers to avoid a fair bet. This is precisely Bernoulli’s explanation to solve St. Petersburg Paradox: people are, in general, risk averse, and hence they are willing to offer a quantity much smaller than the (infinity) expectation value.
Mathematically, there are some properties of implying risk aversion.
Definition 2.4.
A set is convex if for all and , .
Definition 2.5.
Let convex and . A function is concave if for all in its domain,
In the case where , if is twice differentiable it is concave if and only if .
Example 2.6.
The functions and for are concave.
Theorem 2.7 (Jensen’s inequality).
Let be a random variable and be concave and such that . Then
Thus, if the utility function is concave, the investor is (weakly) risk-averse. The converse is also true: risk averse investors must have a concave utility function.
Proposition 2.8.
Assume that is continuous 33 3 i.e., it has continuous density with respect to the Lebesgue measure and that for every random variable ,
Then is concave.
As a consequence of this proposition, at least in the continuous case, there is an equivalence between a risk-averse and a concave utility function. Due to this equivalence, concavity is sometimes directly used as the key property to define risk-aversion.
Exercise.
Show Proposition 2.8
Certainty equivalence
An important tool for investors that decide using their expected utility as a criterion, is to understand what would be a deterministic equivalent of a given random wealth.
Definition 2.9.
Let be a wealth gamble. We denote to be certainty equivalent of :
The name comes form the fact that is a deterministic value that produces the same utility on average than . In view of our assumptions, this value is unique.
Note that an investor is (weakly) risk-averse if for all random wealth . The difference between these two quantities can be understood as a premium that the investor would be willing to pay in order to avoid the uncertainty.
Definition 2.10.
Let be a wealth gamble. The certainty premium44 4 Also known as risk premium in books like Back (2010) of is the (unique) number such that
| (2.3) |
2.3 (Arrow-Pratt) coefficients of risk aversion
Let us fix a given random wealth . Intuitively, the more risk-averse investors are, the bigger the certainty premium they are willing to pay. Thus, the certainty premium can be understood as a measure of risk aversion.
Finding the certainty premium might require numerical techniques 55 5 See for example the Python Notebook 4 on Expected utility . We look in this section for a local approximation to a value that depends only on the derivatives of when they exist.
To do this, let us fix a constant , and consider a small random perturbation , such that
Let us take a small value . We are going to find the certainty premium required on the random endowment as . Clearly, . For simplicity, we write in this section .
Let us suppose that and are two times continuously differentiable on a neighbourhood of , and that (i.e. the utility grows at that point). Applying the Taylor theorem as functions of around , we have on the one hand that
where the notation denotes a value that goes to zero faster than . On the other hand,
By definition, we have equality between the previous two terms. By a scale analysis, we conclude that . Hence, the leading term is quadratic and
Integrating (or replacing on the Taylor expansion), we have that for ,
| (2.4) |
Since represents the variance of , we see that for small perturbations, the certainty premium is linear with respect to the variance, with a coefficient depending only on derivatives of .
2.3.1 Absolute coefficient of risk aversion
Inspired by (2.4), we introduce the following quantity.
Definition 2.11.
The coefficient of “absolute” risk aversion at the wealth level is defined by
| (2.5) |
Therefore, the risk-premium depends (at least as an approximation) on the utility function through the risk aversion, and on the gamble through its variance.
Remark 2.1.
Some observations:
-
1.
The absolute risk aversion is not affected by a monotone affine transform of the utility function.
-
2.
A risk averse investor has an absolute risk aversion .
Exercise.
Justify the observations in Remark 2.1.
2.3.2 Relative risk aversion coefficient
If instead of studying the case , we consider , i.e. if we perturb by a quantity proportional to the actual wealth, and we express the certainty premium in relative terms, i.e. we would have found that
| (2.6) |
This motivates the following definition.
Definition 2.12.
The coefficient of “relative” risk aversion at the wealth level is defined by
| (2.7) |
Remark 2.2.
Relative risk aversion coefficients convey information in the case when the original wealth is strictly positive.
2.4 Notable examples of utility functions
In this section we examine some utility functions commonly examined in the literature.
2.4.1 Constant absolute risk aversion (CARA)
If absolute risk aversion is the same at every level of wealth, that is, , then an investor has CARA utility.
We can identify two cases. If , then
where for to be non-decreasing. This is case is known as linear utility or risk neutral given that . On the other hand, if , we get
| (2.8) |
where is the absolute risk aversion parameter, and must satisfy and . In the case , then and this is a negative exponential utility.
CARA utility implies that risk aversion is insensitive to the total wealth . We will verify that this implies that portfolio choices are independent of the initial wealth.
We can calculate the certainty premium for the CARA utility as follows:
for . Observe that in this case the certainty premium does not depend on , nor but only on .
Example 2.13.
If we assume that then
| (2.9) |
That is, in the case of CARA utility and a normally distributed wealth gamble, the approximation of (2.4) is actually exact for every .
On the other hand, if , then we would have
| (2.10) |
We can choose to retrieve the same variance as before, so this expression shows that the equality does not hold in general.
Example 2.14.
Imagine two investors (oor) and (ich), with wealths and and the same utility function , face the following situation. A coin is tossed, the investors have to pay if tail shows up, otherwise they obtain . Thus, and each with probability and . Since investor has far more money than investor and the game is fair, one would expect that would pay only very little to avoid the game (and much less than ). However, this is not true in the case of CARA utility.
2.4.2 Constant relative risk aversion (CRRA)
Recall the definition of , the coefficient of relative risk aversion, in (2.7). If relative risk aversion is the same at every level of wealth, that is, , then an investor has CRRA utility. Since relative risk aversion is only sensible with positive wealths, we only cosnider the case here . Any CRRA utility function with positive relative risk aversion has decreasing absolute risk aversion
| (2.11) |
Any monotone CRRA utility function is of the form:
-
(i)
;
-
(ii)
;
-
(iii)
.
for , . Cases (ii) & (iii) can be summarised (and rewritten) by
| (2.12) |
Thus, CRRA implies either logarithmic or power utility.
Remark 2.3.
We make a few observations concerning the logarithmic utility:
-
1.
The logarithmic utility has constant relative risk aversion .
-
2.
The logarithmic utility is the limiting case of the power utility for , in the following sense:
(2.13) (Recall that monotone affine transformations do not change the preference structure.)
Exercise.
Prove (2.13). (Hint: L’Hôpital’s rule)
The fraction of wealth that a CRRA investor would pay to avoid a gamble that is proportional to the initial wealth, is independent of the investor’s wealth. To wit, if there is a wealth gamble for some and some with then the risk premium is for some , which does not depend on .
Exercise.
In the case of a CRRA investor, consider a wealth gamble , where is a random variable with and is a constant. Prove that the certainty premium for is of the form , where does not depend on . (Hint: Recall the functional form of a CRRA utility function .)
2.4.3 Hyperbolic absolute risk averse utility functions (HARA)
We say that a utility function belongs to the HARA family, if it has the following form
where , and . Note that the domain where these are defined has to be carefully defined.
HARA utilities encompasses, among other, the CRRA case and the CARA case (the latter as a limit case), and the linear utility case.
2.4.4 Quadratic utility
Whenever and , we get the quadratic utility function. For example, when it reads
| (2.14) |
Note that the utility function is concave and therefore shows risk aversion. However this kind of utility function is not very realistic (an investor would feel very unhappy to have a very large wealth!). Indeed, to make sense of this utility function we need to add the requirement .
Computing expected utility we note that
| (2.15) |
Preferences over wealth gambles therefore depend only on their mean and variance when an investor has quadratic utility.
Exercise.
Show (2.15).
2.5 Utility functions in a multi-period setting
Up to now, we focused on how to understand utility functions in one-period.
We need to consider a generalisation of utility functions that make multiple periods intervene.
We model the fact that investors would like to maximise their perceived expected happiness, via the use of a utility function of their future consumption.
Generalising the case of one dimension we say that a multi-period utility function is a continuous and entry-wise increasing function , with a convex set. It determines an investor’s choices.
We can also generalise the concept of risk-aversion: for any adapted stochastic process in [0,T], and any , we have that
Remark 2.4.
In one period, we could interpret well-being as being wealth or consumption almost indistinctly. In a multi-period setting, there is a difference as one quantity may refer to a stock and the other to a flow. In the above, we interpret the entries of a multi-period utility function as either consumption of additional wealth.
Jensen inequality (this time in a more complicated set) guarantees that concavity (in ) is a sufficient condition for risk aversion.
Given that we have also a time component, it makes sense to understand the behaviour of an investor with respect to time. Most investors show a preference for obtaining their consumption or income sooner rather than later. Let denote the canonical vectors on . We say that a utility function is (weakly) time-discounting if there exists such that for all and all
2.5.1 Time additive utility
A particularly useful family of utility functions in this setting is known as time-additive utility functions. A utility function is time-additive with constant discount factor , if
where is a one-period utility function.
This represents the situation whereby investors are equally satisfied among periods up to a factor representing the preference of having their consumption now. Note that in this case, the increasing and concavity conditions on become the usual ones on .
This type of utility immediately extends the studied families to several periods.
2.6 From rational preferences to utility maximisation(*)
As it seems like a very ad-hoc assumption, several works have been devoted to deduce the utility function maximisation in agents preferences, from simpler assumptions related to rational preference. Here we focus on the one-period case. In Von Neumann and Morgenstern (1945), an axiomatisation was presented that proved an equivalence between the expected utility approach and rational choice.
To describe this set-up, the key element is the set of all probability distributions on the measurable space of real numbers with Borel’s sigma algebra. More general spaces can be used (and indeed need to be used when considering dynamic problems instead of our one-period case). Thus, any wealth profile is actually identified with its probability law, and two wealth profiles with the same distribution are identified. Recall that
In economic literature, these probability distributions are sometimes called “lotteries”.
We consider a binary relation on the space such that is interpreted as “the investment in is (weakly-)preferable to the investment on ”: it means that an investor would not be unhappy if they had to accept the lottery over the lottery . We denote the strong preferences by which means that .
Von-Neumann-Morgenstern propose the following rules that should obey:
Property 2.15 (Von Neumann-Morgenstern).
-
1.
Completeness: For all , either or .
-
2.
Transitivity: If and then .
-
3.
Continuity (or Archimedian): For all , there exist such that if , then66 6 This is well defined, as the convex combination of probability distributions is also a probability distribution. For general measurable spaces a strengthening of the Archimedian property to true continuity of with respect to the weak topology is necessary. See Föllmer and Schied (2011)
-
4.
Independence: For all ,
The stated assumptions can be interpreted in the following way: completeness means that an investor can always decide between any two investments; transitivity means that ordering of preferences are consistent; continuity means that we can replace one investment with a mix of this investment and a third one, provided that the event determining the mix is small enough (it is a rather technical assumption, whose main purpose is to ease the mathematics rather than to answer to a fundamental property). Finally, the independence assumption can be understood as follows: an individual prefers an investment over another if and only if they will still prefer it conditionally on any given event. It is a very strong assumption, and hence it has been debated and modified in the literature.
The main result in Von-Neumann-Morgenstern’s theory is that there exists such a binary relation if and only if an agent acts as if they would like to maximise some expected utility, that is
| (2.16) |
for some (and therefore all) .
Moreover, we can prove that the utility function is unique up to a monotone affine transform. This means that if instead of , an investor decides using a map of the form , where and is considered, then the investor’s preferences remain unchanged.
Further contributions refined these ideas. It is worth mentioning here the work of Savage (1954) who proposed a different set of axioms acting directly over random variables, and that is characterised not just by a concave function but also by a probability function (the subjective view on probability). This approach solves many of the inconsistencies of using the basic utility approach as a descriptive model. Many others, however were still unexplained (see for example Tversky and Kahneman (1992)) where a different approach where utility is defined with respect to a ’reference’ or ’frame’ is introduced. A yet alternative approach was based on introducing uncertainty aversion, that is, on admitting that the probability measure is not fixed but rather that the investors have some doubts and consider a family of distributions around . The alternative probabilities are penalised according to how implausible they seem to the investor. For further reading see Gilboa (2009) and Föllmer and Schied (2011).
2.7 Exercises
Exercise 2.1.
Solve the following two problems.
-
1.
An investor with initial wealth has utility function for some . She faces two options, and decides according to her expected utility. Option 1: doing nothing . Option 2: Betting in a lottery where she might lose 100 or win 100 with equal probability ( or , each with probability ). How does she decide? What about another investor who has utility function ?
-
2.
An investor with initial wealth has utility function and decides between two investments. Investment 1 pays 10 with probability 1/3 and pays 30 with probability 2/3. Investment 2 pays 20 with probability 0.9 and nothing with probability 0.1. Which one does she chose?
Exercise 2.2.
Check that the two functions in Example 2.6 are concave.
Exercise 2.3.
Write a Python program that evaluate the expected utility function if the wealth follows a
-
1.
Pareto distribution with and scale factor .
-
2.
Exponential
-
3.
Log-normal, i.e. with .
Exercise 2.4.
A monotone affine function is an affine map that preserves ordering.
-
•
Show that if is a monotone affine map, then for some and .
-
•
Show that every twice differentiable CARA utility function is a monotone affine transform of the form (2.8).
Exercise 2.5.
Show that if , then the certainty premium for the CARA utility function is given by (2.9).
Exercise 2.6.
What is the absolute risk aversion of a quadratic utility? of a CRRA utility? Are these coefficients increasing or decreasing in their domain?
Exercise 2.7.
Prove that the utility function of (2.12) has constant relative risk aversion
Exercise 2.8.
Consider an investor with relative risk aversion .
-
i.
Verify that the fraction of wealth that he/she is willing to pay to avoid a gamble that is proportional to wealth is independent of the initial wealth.
-
ii.
Consider a gamble . Assume that the gross return of bet is with . What is the average of ? Find the certainty premium that a CRRA investor with initial wealth would pay to avoid betting on .
Exercise 2.9.
This exercise, sourced from Back (2010), is based on a simple bid-ask model by Stoll (1978). Consider an individual with constant risk aversion coefficient . Assume that, if nothing is done, the investor is set to have a wealth at the end of the period of . Consider a pay-off , which is jointly Gaussian with . Assume that has mean , variance and .
-
i.
Compute the maximum amount the individual would pay (at the end of the period) to obtain the pay-off D; that is, compute satisfying
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ii.
Compute the minimum amount the individual would require (at the end of the period) to accept the pay-off ; that is, compute satisfying
(*) Exercise 2.10.
Show that the operator defined over the space of probability functions in such that
for some increasing and concave , satisfies the Von-Neumann-Morgenstern properties listed in Property (2.15)
Exercise 2.11.
Consider an investor with utility.
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i.
Construct a gamble such that
-
ii.
Construct a gamble such that in every state of the world, but Hint: think about the St. Petersburg paradox
Chapter 3 Risk measures
How do we quantify risks? What can be done about them?
We have seen that investors are averse to risk in the sense that they might be willing to pay a premium in return for further certainty in their returns. This is also true in a general sense for companies. There are then incentives for them to understand and manage their (financial) risks. In the case of financial companies this is reinforced by the existence of regulatory requirements for risk management.
Risk management is a system comprising a set of policies, organisational structure, quantitative models and indicators aimed at understanding the risk sources and exposures of a company, deciding and monitoring when they are within what is acceptable, and taking action in cases when they are not. A key element in modern risk management is its quantitative nature. It is based in measuring the risk present in a given investment or in the whole business of a company, so that this can inform actions to reduce it if needed.
The study of a full risk management system and regulation is outside the scope of this lecture. We will mainly focus on how to measure risks. However, the fact that we want to use risk measures to act on them, informs our construction. It is therefore relevant to mention some of the main tools available to companies and investors to reduce financial risks:
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•
Capital (or another form of safe collateral) can be added to the position. Capital is understood as a direct injection of monies, for example, from the owners of a financial company. Moreover, regulators impose capital requirements and restrictions as one of their main tools to promote financial stability.
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•
Hedging, i.e. implementing strategies on the market to counterbalance negative results on a position.
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Diversifying, meaning increasing the heterogeneity of assets or instruments on which there is investment.This is a strategy based on our understanding that there is less variability of results in large numbers.
3.1 Risk measures and their properties
As before, we work on the probability space . We denote by a set of random variables in this space representing all possible financial positions at the end of the period. For example, we might consider the discounted net gains (as defined in (1.7)) from performing any market strategy stating from a fixed amount. We assume that is a convex set (see Definition 3.1 below), and sometimes we assume it is in addition a cone (see Definition 3.1 below). Moreover, we assume that any deterministic value belongs to the cone.
Definition 3.1.
a subset of a vector space is convex if , and it is a cone if for all .
Definition 3.2.
A (unidimensional) risk measure is a function that assigns to a given random variable representing a financial position a real number, representing its “riskiness”.
(!) Note that we use the same letter to denote coefficient of risk aversion and a generic risk measure. The meaning of the symbol should be clear from the context.
Mathematically, a risk measure is just a function: classical examples would include the expectation, standard deviation or worst case (when dealing with finite distributions). As we want to convey a precise financial interpretation we will identify properties that express what we would understand as reasonable risk measures.
To start, it would be intuitively clear that if surely (i.e. under every possible outcome) then represents a riskier financial position.
Property 3.3 (Monotonicity).
For all such that almost surely we have .
Another relevant property is related to our desire to render risk measures useful to determine regulatory capital. Imagine we set the capital to be added to be equal to the measured risk. We would expect that this capital should be sufficient to offset the perceived risk.
Property 3.4 (Cash invariance or translation invariance).
For all and for every , we have .
The monotonicity and cash invariance properties are the minimum properties needed for a risk measure to be used for capital determination.
Definition 3.5.
A risk measure that satisfies both is called monetary risk measure.
Note that any translation of a monetary risk measure is still monetary. For this reason, sometimes the following property is added for convenience
Property 3.6 (Normalisation).
If we want to encode also the fact that diversification reduces risks, we need to add some further properties.
Property 3.7 (Convexity).
For all and then
Convexity simply states that considering a convex combination or a ‘portfolio’ of two financial positions should be less risky than simply combining the measured risks.
Definition 3.8.
A monetary risk measure with the convexity property is called a convex risk measure.
Let us now assume that is also a cone. One can postulate in this case that risks can also be assumed to scale linearly as they grow, i.e.
Property 3.9 (Positive homogeneity).
For all and every we have .
Definition 3.10.
A convex measure that is also positive homogeneous is called coherent risk measure.
Both Positive homogeneity and Convexity together deduce
Property 3.11 (Subadditivity).
For all , we have .
In fact, as pointed out in Föllmer and Schied (2011), we have
Proposition 3.12.
Considering the convexity, positive homogeneity and subadditivity properties, any two of them imply the remaining.
Historically, a first formalisation of the intuitive idea of a what a good risk measure for financial risk management was achieved by Artzner, Delbaen, Eber and Heath in their seminal paper Artzner et al. (1999), where they introduced coherent risk measures (defined as satisfying monotonicity, cash invariance, positive homogeneity and sub-additivity). Then, it was pointed out that the positive homogeneity property might not appropriate in certain contexts, for example when liquidity effects want to be considered. In those cases, one would like for that
This led Föllmer and Schied and Fritelli and Rosazza-Gianin to propose simultaneously the formalisation of convex risk measures.
Finally, let us say that we call a risk measure law invariant if
Let us illustrate the definition an properties of risk measures by reviewing the most classical example of risk measure: standard deviation. Recall that it is defined by
This is one of the first measures to be used in practice due to its simplicity. It is a good measure of dispersion, but as a risk measure, it is quite limited: it is blind to the sign, so it does not distinguish between large losses and large gains. Moreover, it is not defined for all random variables, only for .
Standard deviation can be shown to satisfy homogeneity and subadditivity (thanks to Minkowski’s inequality). Hence it is also convex. However, it is not monotonic or cash invariant (thus not monetary) in general. It is thus not an ideal measure to determine capital requirements.
Let us now look at more general families of risk measures.
3.2 Utility-based risk measures
We can also use the investor preference point of view to define a risk measure.
3.2.1 Simple loss
A first idea to use utility functions applied over losses to determine their risk. More specifically,
Thanks to the monotonicity of , is monotonous and if is concave, is risk measure satisfying convexity. However, it is not in general a monetary risk measure, as cash invariance only holds if is an affine modification of the identity.
3.2.2 Certainty equivalent
A modification of simple losses uses instead their certainty equivalents. Assume that is continuous strictly increasing and concave. Then, we can define
Again, we can easily verify the monotonicity property in the sense of risk measures. The convexity property must be checked on a case by case basis. Moreover, we have the following proposition
Proposition 3.13.
is cash invariant if and only if has constant risk aversion.
Exercise.
Prove Proposition (3.13)
Here is an example of a convex risk measure:
Example 3.14 (Entropic risk measure).
For some , the entropic risk measure with parameter (that we denote ) is defined by
Note that the entropic risk measure is in general not coherent, since for
with a strict inequality for certain distributions (take for example a standard Gaussian and ).
Convexity can be shown using Hölder inequality and the properties of exponentials and logarithms:
3.2.3 Shortfall risk
As we pointed out in Chapter 2, the certainty premium can be seen as a measure of risk. We generalise this idea. Let be a concave utility function. Let denote some fixed utility benchmark. We then define
The shortfall risk associated to the utility is then the minimal deterministic amount that is missing in order to obtain a pre-determined utility.
The shortfall risk here defined is a convex risk measure. It is not in general coherent.
Exercise.
Show that the shortfall risk is a convex risk measure.
3.3 Tail-based risk measures
All the examples of utility functions we have presented up to now either miss the asymmetry between losses and gains, or average out positive and negative results.
To avoid any of these conditions, we can consider measures that focus on the lower tail of the distribution of , where the worst results lie. One way of doing so is by means of a quantile.
Definition 3.15.
The quantile function associated to a random variable , given by
| (3.1) |
where is the cumulative distribution function. It represents the minimal value whose cumulative distribution function is at least .
Remark 3.1.
Note that two random variables that have the same distribution share also the same quantile function. We say that the quantile function is law-invariant.
The quantile function inherits some properties: because the cumulative distribution function is continuous to the right and has a limit to the left11 1 Functions satisfying this property are usually known as càdlàg, a French acronym for continue à droite, limite à gauche , we can show that the quantile function is continuous to the left with limits to the right. In particular, if the CDF is continuous, so it is the quantile function.
A simpler way to characterise (3.1) is as follows:
Proposition 3.16.
The quantile function is the only left continuous with right limits function defined from to such that
| (3.2) |
The proof is immediate from the definition of “infimum” as the minimal value of the set of upper bounds.
In the case where the cumulative distribution function has an inverse in a given domain, we have that from (3.2) that . We can then consider the quantile function as a ‘generalised left continuous inverse’ of the cumulative distribution function .
Let us illustrate with a couple of examples:
Example 3.17.
Let represent the outcome of one fair dice throw. We have that
where the floor function is the largest integer less or equal than . Then, we can find that the quantile function is given by
where the ceil function is the smallest integer greater or equal than . Indeed, note that for any ,
so that (3.2) is verified. This is illustrated graphically in Figure 3.1
 |
 |
Example 3.18.
Let , uniformly distributed. Then we have,
and we find for
as we verify by noticing that
In fact, in this case we have that for and and . As before, we illustrate this graphically in Figure 3.2
 |
 |
The uniform example before that if the cumulative distribution function has an inverse in a given domain, then the quantile function coincides with this inverse on this domain. This confirms that the quantile generalises the inverse of a function.
The following lemma shows another interesting property of the quantile function.
Lemma 3.19.
Let be a random variable with c.d.f. and quantile function . Let be a random variable distributed uniformly in . Then, and have the same distribution.
Proof.
We need to show that the c.d.f. of and that of coincide. Indeed, we have that for any
where the third inequality holds since as a consequence of (3.2), and the last inequality follows from the fact that is uniform in and . ∎
3.3.1 Value at risk
Probably the most popular risk measure is value at risk. Informally, it means to represent the lower value of the ’worst possible losses’ with a given confidence probability.
Definition 3.20 (Value at risk).
Value at risk at level of a wealth , that we denote by , is defined by
If we assume that is a random variable representing profit and losses (P&L), we can interpret value at risk as the minimal value such that when added to our current would guarantee no losses with a probability . Typical values used in market risk management practice are: or .
(!).
Different references have different conventions. Note that the convention here is to calculate the value at risk of a random variable that represents, for example, the profile of results (with positive values meaning profits). The value at risk would have positive value if there is a risk of having negative results. Moreover, has the interpretation of a ’confidence’ level of large losses, so it is typically taken to be close to 1. Some old versions of these notes had a different convention.
Example 3.21.
Assume that a given investment either generates a profit of with probability , or a loss of with probability . What is the value at risk at level of the P& L of this investment?
We call the random variable representing the P&L. Let , representing the losses of this investment. We would have
By following the same procedure as in Example 3.17, we have that
In particular the value at risk is 150 (the maximum loss!) but the value at risk at level is -100 (the negative sign indicates that no losses are expected at that level). This example shows some limitations of as a risk measure.
Example 3.22.
Assume that represents the net profits of a given investment. What is its value at risk at level ?
By following the same procedure as in Example 3.18, we have that
In particular the value at risk is 0.96. So, if we added 0.96 to our initial profit and loss profile, we would not experience losses in of the cases.
Properties of value at risk
is extremely popular: it has a very simple economic interpretation, and it can be, in general easily approximated and tested (we will explore about this in later chapters). It satisfies monotonicity, translation invariance (hence monetary) and the positive homogeneity properties. The proofs are all very similar. Let us show, for instance that is cash invariant.
Proposition 3.23.
is a cash invariant risk measure.
Proof.
From the definition of we have
∎
Value at risk is also a law-invariant risk measure, which is very convenient when applied to market data. However, is not convex or subadditive. Recall that this means that, in general, does not properly recognise diversification as a risk reduction tool.
is, nevertheless, subadditive in some cases where the loss vector , comes from a portfolio composed as linear combination of assets following an elliptical distribution: we define them using their characteristic function. Recall that the characteristic function of a -dimensional distribution is its Fourier transform, i.e, for
and it completely characterises a distribution.
Definition 3.24.
We say that a family of probability distributions in dimension is elliptical if there exists a characteristic function of a scalar variable (called the generator), a vector and a matrix such that
where denotes the probability density function of .
Some examples include the (multivariate) Gaussian, Student, Laplace and logistic distributions.
3.3.2 Expected shortfall
There are some drawbacks of using value at risk. On the one hand, since it focuses on only one quantile, it ignores any information beyond that quantile. In other words it does not inform of how severe the uncommon losses are. Moreover, we remarked that value at risk is not convex. These two shortfalls, together with the eagerness of financial companies to reduce their capital without reducing their average returns, lead to strategies that were barely diversified and carried huge risks with low probability.
Expected shortfall was proposed to deal with these issues. Assuming that is a random variable with finite mean, instead of reporting one quantile which can be seen as the “lower bound of extreme losses”, one can report the average of all the quantiles above a certain margin. This is what is called Expected Shortfall22 2 In this lecture notes we identify expected shortfall, average value at risk and conditional value at risk, which sometimes are distinguished in the literature, particularly for discrete random variables. at level .
Definition 3.25 (Expected shortfall).
For a random variable , expected shortfall at level , denoted is defined by the expression
Note that this definition captures exactly the notion of averaging value at risks at different levels.
(!).
Pay attention to the factor in front of the integral limits of the integral and its sign. In references where a different convention for value at risk is used, they change for consistency.
Other representations
Expected shortfall has other representations that can be very useful to understand its role and properties.
Proposition 3.26.
Let . Then,
| (3.3) |
Proof.
We have that
for a random variable on . Now, let be given as in the statement of the proposition. From (3.1), it is clear that . We can then write
Using the definition of the expectation conditional to the event we immediately deduce
Corollary 3.27.
Let where is defined as in Proposition 3.26. Then
| (3.4) |
Clearly, the second term in both (3.3) and (3.4) vanishes when (for example if the distribution has an inverse).
Corollary 3.28 below expresses expected shortfall in yet another way. It can be interpreted as adding the expected value plus the expected value of a put option with strike minus the expected value.
Corollary 3.28.
| (3.5) |
where , , and .
In fact we can strengthen Corollary 3.28 as follows
Proposition 3.29.
The proof is based on the fact that the function to minimise is convex (so every local optimal is global) and can be worked out by perturbing the proposed optimal argument , and then concluding from Corollary 3.28.
Exercise.
Show Proposition 3.29.
Examples
Example 3.30.
If we calculate value at risk with the data of example (3.21), we get that
For example, we get that , and .
Example 3.31.
If we calculate value at risk with the data of example (3.22), we get that
In particular, we get that .
is a coherent (and hence also a convex) risk measure. It is also very popular, and has progressively replaced as the de facto risk measure for risk management.
The main drawbacks of expected shortfall are that it can only applied to random variables with finite mean (which exclude some heavy tail distributions), it is more difficult to estimate and to validate than .
Properties
It is easy to show that the fact that expected shortfall inherits the monotonicity, cash invariance and positive homogeneity properties from value at risk, thanks to the monotonicity and linearity properties of the integral. Expected shortfall is also law-invariant. However, expected shortfall is in addition subadditive (and thus convex).
Proposition 3.32.
is subadditive.
Proof.
3.4 Summary of properties
| Monotonicity | |||||
| Translation invariance | |||||
| Subadditivity | |||||
| Positive homogeneity | |||||
| Convexity | |||||
| Normalisation |
Variance and standard deviation are monotonous when restricting to losses with the same mean.
is subadditive (and hence convex) when considering linear combinations of a multidimensional elliptic function.
3.5 Robust representation(*)
Coherent and convex risk measures have a robust representation property in the sense that they can be expressed in terms that do not require a probability to be fixed, provided that they satisfy the following regularity result.
Definition 3.33 (Continuity from above).
A risk measure is continuous from above if for any sequence of random variables such that implies .
Let be the set of probability measures such that for all , is well defined for all . We have the following robust representation (see for example Föllmer and Schied (2011)).
Theorem 3.34.
Assume that is a coherent risk measure continuous from above. Then, there exists a set such that
| (3.6) |
Moreover, can be chosen as a convex set for which the supremum is attained.
Intuitively, the theorem implies that a coherent risk measure takes into account uncertainty of the probability model of the possible outcomes: its value is the worst average loss that can be obtained amongst all the possible distributions in the set .
As an example, assuming that , we get
where
| (3.7) |
Convex risk measures
In the case of a convex risk measure, a similar representation exists. This time there is an extra penalisation term.
Theorem 3.35.
Let be a convex risk measure continuous from above. Then there exist a set and a function such that
| (3.8) |
Furthermore, defining
Loosely speaking, the risk measure is once again an average on many possible measures on the set , but where we have penalised (through the term ) probability distributions that are considered “unrealistic”.
3.6 Risk mitigation
A notion that is frequently used in risk management is that of acceptable positions. Mathematically, we model these positions as a subset of the set of all possible economic positions . Investors and companies will avoid any position that falls outside this set. Risk mitigation is the process of using financial tools to render an economic position acceptable.
A simple way to determine an acceptability set is based on risk measures
Definition 3.36.
The acceptability set associated to a risk measure is defined by
A position is acceptable if .
In the following we examine some tools of risk mitigation under this framework.
3.6.1 Collateral addition
Collateral is understood as a financial deposit, usually required to be given in terms of a very secure asset, provided to mitigate losses associated to some type of default. Note that the deposit is not used unless the risk is materialised.
The amount of collateral required to render a position acceptable depends on the numeraire on which it is provided:
-
•
If collateral is provided in a risk-less numéraire and is monetary , that is the amount if it is positive (which in practice is usually the case).
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•
If collateral is deposited using a defaultable numéraire ,
If is strictly convex , there is at most a unique solution. If the solution is infinity (or infinity in practice) the numéraire is not adequate for collateral provision.
Capital as a form of collateral
One of the pillars of financial regulation is the need to provide capital to support any risky operation, as a way to reduce the likeliness of financial crises. In practice, financial regulation also establishes the types of instruments in which capital can be provided (for example AAA sovereign bonds with maturity 30 years or corporate AA+ bonds with maturity 10 years) and in which proportions (for example 70% and 30% respectively). See the notes and the end of the chapter for more information on capital regulation.
3.6.2 Hedging
Another way to mitigate risks is to hedge those risk, that is, to make a sequence of operations in the market to obtain pay-offs that offset the initial risks.
With the convention that denotes discounted net position, that is for some pay-off at final time, we look for a strategy with the minimal initial cost such that , i.e.
If such a exists, we say that the strategy partially hedges . Of course, if there is a replication strategy i.e.
the set on the right hand side is not empty, but it can be seen that lower initial strategy costs are obtained if we limit ourselves to partial hedging.
3.7 Exercises
Exercise 3.1.
Assume that and . Check that
where denotes the standard Normal CDF.
Exercise 3.2.
Assume that and . Check that
where is the standard Normal PDF and denotes the standard Normal CDF.
Exercise 3.3.
Compute and when is
-
•
uniform
-
•
log-normally distributed; that is, , where is standard normally distributed and .
Exercise 3.4.
Give an example of two different distributions that have the same value at risk and expected shortfall at a level. Can you define the example in such a way that one of the distributions would pose a bigger threat to the survival of a company? Justify your answer.
Exercise 3.5.
Consider a setup with defaultable bonds. Each bond has price . Assume that defaults are independent and each default probability equals . If there is a default, the bond pays , otherwise .
Consider two portfolios. Portfolio A consists of 100 copies of bond ’1’, Portfolio B consists of one unit of each of the bonds.
Compute for each Portfolio A and Portfolio B (For the second computation, you may use any computer language, and find the correct value by trial-and-error). What do you observe?
Exercise 3.6.
Give an alternative proof for (3.3) in the case where has a strictly positive probability density function . Remark: Note that this case, .
Exercise 3.7.
Show that the negative expectation is a coherent risk measure 33 3 In fact, as observed in Artzner et al. (1999) all coherent measures are of this form, up to a change of probability measure. .
Exercise 3.8.
Suppose that the probability space is finite, i.e. Show that the worst-case risk measure is a coherent risk measure by verifying its properties. Find the set associated to the robust representation of this measure.
Exercise 3.9.
Assume that is a convex risk measure. (This is, satisfies monotonicity, cash invariance, and convexity.) Assume also that is normalised; that is, . Show that
Exercise 3.10.
Consider buying a risky asset with price at the beginning of the period. Assume its price at the end of the period “” is log-normally distributed with
How much capital (in riskless numéraire and assuming no interest rate), would be needed to cover the risk of the profits or losses () of the operation, when using:
-
1.
-
2.
Exercise 3.11.
Assume you already own the risky asset of exercise 3.10. Assume that it is possible to buy any nominal value of
-
•
A call option with strike and price , that is an asset with pay-off
-
•
A put option with strike and price , that is an asset with pay-off .
Can you partial hedge, replicate and super hedge your current risk by buying one of those assets? Justify your answers giving the quantities of each option needed.
(*) Exercise 3.12.
Let be a coherent risk measure. Use the robust representation to show that is additive, i.e.,
if and only if the class reduces to a single probability measure , i.e., is simply the expected loss with respect to .
(*) Exercise 3.13.
Let denote a sequence of independent and identically distributed random variables such that . Show that
(Hint: Assume first that , and use the law of large numbers.)
3.8 Notes
On risk measures
Let us remark that as was shown in Exercise (3.4), there might be blind spots, for a risk measure that is, risky positions that might potentially harm the subsistence of a company, but that do not appear on a risk measure calculation.
There are many additional interesting properties that risk measures can have (ellicitability, robustness, …), some of which we will examine when considering applications. The desire to have as many of this properties as possible has lead to many papers proposing new types of families of risk measures, but they sometimes sacrifice intuitive understanding. Some well-known families include expectiles and distortion risk measures.
On risk management and mitigation
The best source to learn about financial risk management regulation is to go to the website of the local regulator (The Bank of England, for example in the UK). However, an interesting overview of the regulation framework can be found on the documents of the Basel Committee on Banking Supervision (BCBS). The BCBS describes its objective as to “enhance understanding of key supervisory issues and improve the quality of banking supervision worldwide”. It is a joint effort of regulators of 28 countries to coordinate and improve banking/financial regulation, functioning more as a respected but informal forum on how to make regulation converge.
The most important contribution of the Basel Committee is the construction of the so-called Basel Accords, frameworks that state the general guidelines and principles that local regulators later develop. The most recent one, Basel III developed between 2010 and 2011, was introduced in the aftermath of the last financial crisis. Of particular importance is the chapter denoted Fundamental Review of the Trading Book.
Very schematically, the current paradigm for regulation(see for example Basel Committee on Banking Supervision (2016)) allows for two approaches for capital calculation:
-
•
The standardised approach for capital requirements is based on the linearisation-normal approximation. The idea is as follows: financial companies calculate sensitivities and exposures and the regulation provides the coefficients associated to them.
-
•
In the internal model approach, an institution proposes a model for capital calculation (which, as shown, includes calculating risk measures) that is adjusted to their business, theoretically sound and empirically shown to work (not fail, in fact). It is subject to the approval of the regulator.
The agreement documents and the discussions to modify it on time are available and even open to comment on the Committees web-site: https://www.bis.org/bcbs/
Bibliography
- Coherent Measures of Risk. Mathematical Finance 9 (3), pp. 203–228. External Links: ISSN 14679965, Document Cited by: §3.1, footnote 3.
- Asset pricing and portfolio choice theory. Vol. Financial Management Association survey and synthesis series, Oxford University Press, New York, New York. External Links: ISBN 0-19-538061-4 Cited by: Exercise 2.9, footnote 4, 1st item.
- Minimum capital requirements for market risk. Technical report Bank for international settlements. Cited by: §3.8.
- Exposition of a New Theory on the Measurement of Risk. Econometrica 22 (1), pp. 23–36. External Links: ISSN 0012-9682, Document Cited by: footnote 2.
- Arbitrage theory in continuous time. Oxford university press. Cited by: 3rd item.
- Convex optimization. Cambridge University Press, Cambridge, UK ; New York. External Links: ISBN 978-0-521-83378-3, LCCN QA402.5 .B69 2004 Cited by: 4th item.
- Probability: Theory and Examples Rick Durrett January 29, 2010. Biometrics 49 (3), pp. 497. External Links: ISSN 0006341X, Document Cited by: §1.5, 1st item.
- Stochastic finance: an introduction in discrete time. Walter de Gruyter. Cited by: §2.6, footnote 6, §3.1, §3.5, 2nd item.
- Theory of decision under uncertainty. Econometric Society Monographs, Vol. 45, Cambridge University Press, Cambridge. External Links: ISBN 978-0-521-74123-1, MathReview Entry Cited by: §2.6.
- Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach. 2 edition edition, Prentice Hall, Upper Saddle River, NJ (English). External Links: ISBN 978-0-13-041408-3 Cited by: 3rd item.
- Methods of mathematical finance. Vol. 39, Springer Science & Business Media. Cited by: 3rd item.
- Linear algebra. 3. ed., [softcover repr.] edition, Undergraduate Texts in Mathematics, Springer, New York, NY (eng). Note: OCLC: 837651207 External Links: ISBN 978-1-4419-3081-1 Cited by: 2nd item.
- Quantitative risk management: Concepts, techniques and tools. Princeton university press. Cited by: 4th item.
- Convex analysis. Princeton university press. Cited by: Theorem 1.36.
- The foundations of statistics. John Wiley & Sons, Inc., New York; Chapman & Hill, Ltd., London. External Links: MathReview Entry Cited by: §2.6.
- Calculus on manifolds: a modern approach to classical theorems of advanced calculus. 27. print edition, Mathematics Monograph Series, Perseus Books, Cambridge, Mass (eng). Note: OCLC: 176892168 External Links: ISBN 978-0-8053-9021-6 Cited by: 3rd item.
- Advances in prospect theory: Cumulative representation of uncertainty. Journal of Risk and uncertainty 5 (4), pp. 297–323. Cited by: §2.6.
- Theory of games and economic behavior. Bull. Amer. Math. Soc 51 (7), pp. 498–504. Cited by: §2.6.
- Probability with martingales. Cambridge Mathematical Textbooks, Cambridge Univ. Pr, Cambridge (eng). Note: OCLC: 930941220 External Links: ISBN 978-0-521-40605-5 Cited by: 1st item.